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A046951 a(n) is the number of squares dividing n. 52
1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Rediscovered by the HR automatic theory formation program.

a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001

We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009

Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014

Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

A. Amariti, C. Klare, D. Orlando, S. Reffert, The M-theory origin of global properties of gauge theories, arXiv preprint arXiv:1507.04743 [hep-th], 2015 (see (A.13)).

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

S. Colton, HR - Automatic Theory Formation in Pure Mathematics

I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).

A. V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.

W. G. Nowak and L. Tóth, On the average number of subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1307.1414 [math.NT], 2013.

N. J. A. Sloane, Transforms

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016

a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.

Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007

a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007

Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).

G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002

a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013

a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014

From Antti Karttunen, Nov 14 2016: (Start)

a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).

a(n) = A278161(A156552(n)).

(End)

G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016

EXAMPLE

a(16) = 3 because 1*16 = 16 and 1|16, 2*8 = 16 and 2|8, 4*4 = 16 and 4|4.

G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

MAPLE

A046951 := proc(n)

    local a, s;

    a := 1 ;

    for p in ifactors(n)[2] do

        a := a*(1+floor(op(2, p)/2)) ;

    end do:

    a ;

end proc: # R. J. Mathar, Sep 17 2012

MATHEMATICA

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)

Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)

a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)

a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)

PROG

(PARI) a(n)=my(f=factor(n)); for(i=1, #f[, 1], f[i, 2]\=2); numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

(PARI) a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

(PARI) a(n)=factorback(apply(e->e\2+1, factor(n)[, 2])) \\ Charles R Greathouse IV, Sep 17 2015

(Haskell)

a046951 = sum . map a010052 . a027750_row

-- Reinhard Zumkeller, Dec 16 2013

(Scheme)

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))

(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))

;; Antti Karttunen, Nov 14 2016

CROSSREFS

Cf. A000005, A000188, A004101, A005117 (positions of ones), A008619, A038538, A046952, A052304, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A156552, A278161.

Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).

Sequence in context: A185102 A049419 A159631 * A050377 A255231 A279848

Adjacent sequences:  A046948 A046949 A046950 * A046952 A046953 A046954

KEYWORD

nice,nonn,mult

AUTHOR

Simon Colton (simonco(AT)cs.york.ac.uk)

EXTENSIONS

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

STATUS

approved

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Last modified May 28 10:09 EDT 2017. Contains 287240 sequences.